StudyGuide
DepartmentofCivilEngineering

SWK 122 Mechanics

Last revision July 2020

© Copyright reserved


Contents
ORGANISATIONAL COMPONENT 1

1 Introduction 1

2 Lecturers 1

3 Textbook 2

4 Learning activities 2
4.1 Lectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4.2 Tutorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4.3 Timetable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4.5 Learning hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4.6 Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

5 Rules of assessment 3
5.1 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
5.2 Marked tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.3 Correction of a student record . . . . . . . . . . . . . . . . . . . . . . . 5
5.4 Semester mark and final mark . . . . . . . . . . . . . . . . . . . . . . . 5

6 Module requirements 6
6.1 Lecture and tutorial class participation . . . . . . . . . . . . . . . . . . 6
6.2 Admission to the examination . . . . . . . . . . . . . . . . . . . . . . . 6
6.3 Pass requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6.4 Supplementary examination . . . . . . . . . . . . . . . . . . . . . . . . 6

7 Writing tests and examinations 6

7.1 Extra time for class tests, semester tests and examinations . . . . . . . 7

8 Absence from tests and examinations 8

8.1 Absence from class tests and semester tests . . . . . . . . . . . . . . . 8
8.1.1 Absence due to illness . . . . . . . . . . . . . . . . . . . . . . . 8
8.1.2 Absence due to extraordinary circumstance . . . . . . . . . . . 9
8.2 Absence from the examination . . . . . . . . . . . . . . . . . . . . . . 9

9 Sick/extraordinary tests 10

10 Disciplinary cases 10

11 Grievances procedure 10

12 Student support 11

13 General 11
 13.1 Announcements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
13.2 Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13.3 ClickUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13.4 Previous tests and examination papers . . . . . . . . . . . . . . . . . . 12
13.5 Emails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13.6 Writing the examination or the supplementary examination without
qualifying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13.7 Perusal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13.8 Bargaining for marks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
13.9 Formula page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

STUDY COMPONENT 14

1 Use of the study guide 14

2 General objectives 15

3 Pre-knowledge 15

4 Articulation with other modules in the programme 15

5 Critical learning outcomes 16

6 Module structure 16

Theme 1 17

Theme 2 27

Theme 3 36

Theme 4 38

Theme 5 42

SUPPLEMENTARY INFORMATION 44

Study guide answers 45

Textbook answers 48

clickUP Evaluation 50

Centroids formula sheet 53

Special note: Due to the current lockdown restrictions, this study guide is written
under the assumption that the delivery of the course will be online for the entire
semester. Should it at any point become possible for in-person delivery of lectures,
tutorials and/or assessments, updated direction will be provided via clickUP and a
revised study guide provided if necessary.

ORGANISATIONAL COMPONENT
1 INTRODUCTION
This module acts as an introductory course to the statics of rigid bodies. The main
topics covered include equilibrium of particles and bodies in two and three dimensions,
structural analysis of trusses and frameworks, centroids and second moments of area,
and internal forces in structural members (beams).

The prerequisite for entry to the module is WTW 158/114 GS (final mark of
40% or entrance into a supplementary examination) Note: in light of the Covid 19
pandemic, a decision was made to relax the prerequisite from WTW 158/114 pass to
WTW 158/114 GS for S2 2020.

2 LECTURERS
Module coordinator Dr A Roux
Lecturers
Prof WP Boshoff
Dr TS Burke
Dr A Roux
Ms MS Smit

Consultations & queries

Due to the current lockdown restrictions, there will be no contact consultation hours.
The lecturers and tutors are only available for online consultation during the tutorial
and scheduled question & answer sessions (see section 4 for further details). This
arrangement remains in place before tests and the examination. This policy aims
to encourage students to plan their work and to work continuously throughout the
semester.

Students are encouraged to post queries related to course material on the relevant
discussion boards in clickUP; queries posted here will be addressed in the tutorials
and question & answer sessions. This allows all students taking the course to benefit
from seeing the explanations for common problems.

The procedure to contact the lecturer team will be posted in the SWK122 clickUP
homepage. Only queries sent via the indicated procedure will be dealt with; emails
sent to the lecturers’ personal email addresses will not be attended to.

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3 TEXTBOOK
Engineering Mechanics: Statics, Fourteenth Edition in SI Units.

Author: R C Hibbeler.

Publisher: Pearson.

Language: English.

ISBN: 978-1-292-08923-2.

Available to buy as an e-book: https://shop.pearson.co.za/9781292089294

4 LEARNING ACTIVITIES
4.1 Lectures
Number of lectures per week: 4 (50 minutes / lecture).

All lectures will be delivered as pre-recorded online lectures available for viewing
and/or download on clickUP. Course material will be made available weekly; these
recordings will stay available for the whole semester and you may go back to previous
lectures when you run into problems with follow-up lectures.

There are three interactive question and answer sessions and two semester
test discussions scheduled as live collaborate sessions during the lecture times;
please consult the course schedule on the SWK122 clickUP homepage for further
information. These are an opportunity for students to address queries with the
course material with the lecturers. The live collaborate sessions are not available via
the UP Connect zero rated data portal; these sessions will be recorded, and these
recordings will be available via this portal.

4.2 Tutorials
Number of tutorials per week: 1 (100 minutes / tutorial).

The tutorials will be delivered as live collaborate sessions, facilitated by the tutor
team. Students can ask questions live during the session, or use the clickUP discussion
board to post queries to be addressed prior to the session. The live tutorial sessions
are not available via the UP Connect zero rated data portal; these sessions will be
recorded and uploaded afterwards, and these recordings will be available via this
portal. Please attend the tutorial session as scheduled in your timetable to assist in
managing student numbers in the online sessions.

4.3 Timetable
The lecture and tutorial timetable is published by the University Administration.
The schedule of lectures and tutorials for this module is available on clickUP.

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4.4 Objective
The main objective of the module is to develop the skills of the student in formu-
lating mathematical models for given mechanical situations, solving the models and
interpreting the results. Like any other skill, this can only develop and be achieved
through practice. This practice is obtained by understanding and proving the under-
lying mathematics, as well as solving problems. It is of critical importance that the
student solves the problems himself/herself and does not obtain the solutions from
any other source.

4.5 Learning hours

This module carries a weight of 16 credits, indicating that on average a student
should spend approximately 160 hours on the module to master the required skills
(including time for preparation for tests and examinations). This means that on
average you should devote 12 hours of study time per week to this module. The
scheduled lecture and tutorial time is 5 hours per week, leaving you with another 7
hours per week of own study time. The actual time required to complete the module
successfully, depends on the abilities and circumstances of each student.

You may not have enough time to do all the problems in the textbook. At the end
of each lecture unit, typical problems are listed. Attempt these problems first, and
if you are of the opinion that you need more practice, do extra problems. Additional
problems may be given during lectures and tutorials.

4.6 Progress
It is the student’s responsibility to ensure that he/she reviews all the provided ma-
terial and keeps up to date with the work, as set out in the schedule on clickUP.
Students are encouraged to work through the course material in their own time or
during the allocate lecture slots in the timetable. Student’s progress will be monitored
through the scheduled evaluations as discussed in point 5.1.

5 RULES OF ASSESSMENT
5.1 Evaluation
The module evaluation will be based on electronic assignments and tests submitted
via clickUP. Read the general instructions on clickUP evaluation in the back of the
study guide.

The evaluation process will be discussed in the first pre-recorded lecture on 3 and 4
August 2020 (depending on your personal timetable) and a PDF copy of this lecture
will be made available on clickUP.

The complete evaluation schedule is posted on the GENERAL INFORMATION

page on clickUP and the applicable week schedule will be updated as a separate
paragraph for your convenience.

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The evaluation consists of the six activities (a) lecture assignments (LA), (b) tutorial
preparations (TP), (c) tutorial worksheets (TW), (d) class tests, (e) semester tests
and (f) the examination. A description of each of these activities follows.

(a) Lecture assignments (LA)

(i) The lecture assignments assess fundamental concepts and theory

per lecture unit and serve as preparation for the pre-recorded
lecture.
(ii) All the LA marks contribute to the semester mark.

(b) Tutorial preparation (TP)

(i) For each tutorial, part of the preparation will be done via clickUP.
(ii) The tutorial preparations assess basic principles and problems for
the upcoming tutorial.
(iii) All the TP marks contribute to the semester mark.

(c) Tutorial worksheets (TW)

(i) A tutorial worksheet will be uploaded at the end of each lecture week for
the next week. You can complete the worksheet during your assigned tuto-
rial time slot on the timetable. There will be lecturers/tutors available via
collaborate sessions for consultation during the formal tutorial timeslots.
(ii) After completion of the relevant TP, you complete the worksheet by hand.
If you have sorted out all your problems, you access the clickUP worksheet
(TW), where you get the same generic questions as in the worksheet, but
with a different set of parameters.
(iii) The tutorial worksheets assess the understanding of more complex
problems and serve partly as preparation for class tests, semester
tests and the examination.
(iv) All the TW marks contribute to the semester mark.

(d) Class tests

(i) Three clickUP class tests are scheduled for the following dates and times.
Monday 2020-08-24 (Monday 24 August) 17:30 - 18:30
Monday 2020-10-05 (Monday 5 October) 17:30 - 18:30
Monday 2020-11-02 (Monday 2 November) 17:30 - 18:30
(ii) These tests assess more complex problems and serve as preparation
for semester tests and the examination.
(iii) All the class test marks contribute to the semester mark.
(iv) Material for tests will be published on clickUP.

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 (v) A mock test is scheduled a week prior to the first class test to allow students
to test the procedures in place for online tests. It is essential that all
students participate in this mock test; any reports of problems accessing
further tests and exams will be checked against participation in the mock
test.
Monday 2020-08-17 (Monday 17 August) 17:30 - 18:30

(e) Semester tests

(i) Two semester tests are scheduled in the two semester test weeks. The
semester test dates will be published on clickUP.
(ii) These tests asses fundamentals, theory and complex problems and
are in general of a much higher standard than the LA and TP evalua-
tions.
(iii) Both semester tests contribute to the semester mark.
(iv) Material for tests will be published on clickUP.

(f) Examination

(i) The examination is scheduled for the November examination period.

(ii) The examination assesses fundamentals, theory and complex prob-
lems and are in general of a much higher standard than the LA and
TP evaluations.
(iii) The examination and supplementary examination dates are de-
termined by the faculty administration and the central roster committee.
The lecturers and the Department of Civil Engineering are not involved
in any way in setting these dates, and the dates cannot be changed.

5.2 Marked tests

Save the Summary Sheets for all your clickUP evaluations. They are your only proof
that you have written the tests. Keep a file with your handwritten calculations for
all your evaluations; these may be requested as proof in the event of discrepancies in
marks or evaluations of irregularities.

5.3 Correction of a student record

If there is a problem with your record, you must submit the summary sheets and
hand calculations for the relevant evaluation(s) to the lecturer team using the query
procedure available on clickUP.

5.4 Semester mark and final mark

The semester mark and the final mark will be composed as follows:

5
 Semester mark Final mark
Semester test 1 30%
Semester test 2 45% Semester mark 50%
Class tests 15% Examination mark 50%
Continuous evaluation (LA, TP & TW) 10%

6 MODULE REQUIREMENTS
6.1 Lecture and tutorial class participation
It is compulsory to watch all the pre-recorded lecture videos and to either attend the
live tutorial sessions or watch the saved recordings. This is also applicable even for
students who are repeating the module.

6.2 Admission to the examination

A semester mark of at least 40% is required for admission to the examination.

6.3 Pass requirements

A final mark of at least 50% and a sub-minimum of 40% for the examination is
required to pass the module.

6.4 Supplementary examination

• A student qualifies for a supplementary examination if one of the following
three conditions is met on the final mark:

o The final mark is between 45% and 49%.

o The final mark is between 40% and 44% and either the semester mark or
the examination mark is at least 50%.
o The final mark is at least 50% but the required sub-minimum of 40% for
the examination has not been obtained.

• The semester mark and supplementary examination mark each contribute 50%
towards the final mark.

• A sub-minimum of 40% for the supplementary examination and a final mark

of 50% is required to pass the module.

• The maximum final mark is 50%. If the average of your semester mark and
supplementary examination is more than 50%, your final mark becomes 50%.

7 WRITING TESTS AND EXAMINATIONS

It is assumed that all students are aware of all regulations as published by the Univer-
sity and Faculty. Timetables of test weeks (for semester test 1 and 2), the examination

6
 and supplementary examination are set and managed by EBIT and the central roster
committee. These dates will be published on clickUP.

Take into account the possibility that you may have to write a supplementary ex-
amination. No special arrangements with regard to the supplementary examination
date will be made.

• General regulations and rules

Familiarise yourself with the content of the general rules with regard to this
matter. It can be found in the General Regulations and Rules booklet, in
the section with heading 2.2 Examinations, Tests and other Academic
Assignments.

• Plagiarism
Plagiarism is a serious form of academic misconduct. It involves both appro-
priating someone else’s work and passing it off as one’s own work afterwards.
Thus, you commit plagiarism when you present someone else’s written or cre-
ative work (words, images, ideas, opinions, discoveries, artwork, music, record-
ings, computer-generated work, etc.) as your own. Only hand in your own
original work. This means that during tests and exams, you may not
consult any one else.
For more details visit visit the library’s website:
http://www.library.up.ac.za/plagiarism/index.htm.

• Ethical Conduct Statement

When completing the online class tests, semester tests and examinations, stu-
dents will have to agree to a declaration on ethical conduct, which states:
“The University of Pretoria commits itself to produce academic work of in-
tegrity. I affirm that I am aware of and have read the Rules and Policies of the
University, more specifically the Disciplinary Procedure and the Tests and Ex-
aminations Rules, which prohibit any unethical, dishonest or improper conduct
during tests, assignments, examinations and/or any other forms of assessment.
I am aware that no student or any other person may assist or attempt to assist
another student, or obtain help, or attempt to obtain help from another stu-
dent or any other person during tests, assessments, assignments, examinations
and/or any other forms of assessment.”

7.1 Extra time for class tests, semester tests and examinations
• The only valid and applicable document for a student who applies for ex-
tra time, is a letter from the Disability Unit of the University of Pretoria.
No other letter will be accepted.

• It is the responsibility of the student to inform the lecturer team using the
procedure indicated in clickUP in due time of such an application before your

7
 class tests, semester tests and examination. A list for extra time students is
updated as the information comes in. If your name has been added to the list,
you do not have to inform us again for tests to follow. If your name is not on
the list, no provision for extra time for you will be made.
• Extra time does not apply for LAs, TPs and TWs, as ample time is given for
these evaluations.
• Information regarding the organisation of online class tests, semester tests and
exams for extra time students will be communicated via clickUP prior to the
test or exam. It is the responsibility of the student to make sure of the special
arrangements for extra time.

8 ABSENCE FROM TESTS AND EXAMINATIONS

Any absence from a test or examination must be supported by an official and valid
statement. The notification procedures are different for tests and examinations; these
are detailed below.

The submission of the required documentation (i) does not necessarily give an
absent code in the case of a class test, (ii) does not automatically give admis-
sion to a sick/extraordinary test when absent for a semester test, and (iii) does
not automatically give admission to a supplementary examination when
absent for the examination. Your submission will be evaluated on merit.

8.1 Absence from class tests and semester tests

The lecturer team must be notified of the absence with a valid reason from class
tests and semester tests only. This must be done as soon as possible from the
date of the test. The notification procedure is provided on the SWK122 clickUP
homepage.

The procedure for submitting documentation is described in points 8.1.1 and 8.1.2 for
absences due to illness and extraordinary circumstances respectively. The procedure
may change if lock down restrictions change.

8.1.1 Absence due to illness

Please complete the following form for absences due to illness:

https://forms.gle/Zu9qk1bWZEJVdf3dA

This must be completed within three working days from the date of the test.

Please note the following:

(i) Keep your original medical certificate and other documentation safe for further
reference, if needed.

8
 (ii) Do not send the documentation to the secretary of civil engineering or to any
personnel member in the civil engineering department.

(iii) If you are a student from another department, please submit the relevant
documentation as described above, even if you have submitted to your depart-
ment.

(iv) A medical certificate stating that a student appeared ill or declared himself/
herself unfit to write a class test or a semester test, will not be accepted.

(v) The doctor must be consulted on or before the scheduled test date.

(vi) The original certificate must comply with the rules as described in Ethical
Rules of Conduct 2016 of the Health Professions Council of South Africa with
respect to the paragraphs information on professional stationery (section 4)
and certificates (section 16).

Points (iv) and (v) are in accordance to the Regulations for the degree: BEng, ENG 3,
Examinations, (f ) Special examinations, in the Engineering Yearbook.

8.1.2 Absence due to extraordinary circumstance

Please complete the following form for absences due to extraordinary circumstances:
https://forms.gle/ubg8QG7pEqA9LidV9

It is the student’s responsibility to keep a detailed record of valid proof for absences
due to extraordinary circumstances.

Extraordinary circumstances due to any internet connection and/or power supply

problems need to be sent to the lecture team as soon as possible and within a
maximum of six hours from the starting time of the test for these to be considered.
A dedicated email address to be contacted during tests and exams will be indicated
on the SWK122 clickUP home page and in the test instructions.

Extraordinary circumstances due to any other circumstance must be completed within

three working days from the date of the test.

8.2 Absence from the examination

The examination is a faculty administration matter. The student informs the relevant
faculty administration directly of his/her absence of the examination; NAS students
inform the NAS faculty and EBIT students inform the EBIT faculty.

The same rules regarding the validity of the absence, documentation to be completed,
and timeframes for submitting the notice apply as in Sections 8.1.1 and 8.1.2.

9
9 SICK/EXTRAORDINARY TESTS
Sick/extraordinary tests for class tests:

There is no sick/extraordinary test for absence from any of the class tests. Absence
from a class test with a valid reason (see point 8.1) will be taken in account when
calculating the final class test mark contribution to the semester mark.

Sick/extraordinary test for semester tests:

• There is one sick/extraordinary test for absence from any one of the two
semester tests. This test will be conducted after the second semester test.

• The date and time will be announced on clickUP and the responsibility lies
with the student to make sure of the information.

• The sick/extraordinary test covers all the material for both semester tests.

• This test is compulsory for a student who was absent with a valid reason from
the missed semester test.

• The sick/extraordinary test is not an opportunity to better your marks. Do

not approach the lecturers with such a request; the answer will be categorically
no.

10 DISCIPLINARY CASES
The policy of the Department of Civil Engineering is to refer all cases where even
the slightest suspicion of irregularity exists, without exception to the Disciplinary
Committee of the university.

11 GRIEVANCES PROCEDURE
All grievances must be submitted in writing with specifics of the incident or the
nature of the complaint. It is imperative that you follow the procedure outlined
below in order to resolve your issues:

1. It is the student’s responsibility to keep up with the work following the schedule
on clickUP. If there are any grievances/concerns, contact the lecturer team via
the procedure indicated on the clickUP homepage.

2. If the matter has not been resolved, proceed through the following steps in
chronological order.

(i) Consult the class representative. (The primary function of the class rep-
resentative is to serve as a two-way communication channel between the
class and the lecturer.)

10
 (ii) Consult the module coordinator if the matter has not been resolved by the
class representative.
(iii) Consult the Head of Department (HOD) if the matter has not been re-
solved by the module coordinator.
(iv) Consult the Dean of the Faculty if the matter has not been resolved by
the HOD.

12 STUDENT SUPPORT
The University of Pretoria supports you in various ways free of charge.

Academic support

Contact the lecturers allocated to the module, and/or the Faculty Student Advisor.
https://www.up.ac.za/teaching-and-learning/article/2494904/faculty-student-advisors-
fsas

Academic support.
Faculty Goal setting & motivation. Individual consultations
student Adjustment to university life. and workshops about
advisers Test / Exam preparation. − time management
Stress management. − study methods
Career exploration.

• Think carefully before dropping modules (after the

FLY@UP: closing date for amendments or cancellation of
modules).
The Finish • Make responsible choices with your time and
Line is work consistently.
Yours • Aim for a good semester mark.
• Don’t rely on the examination to pass.
• Visit www.up.ac.za/fly@up or email fly@up.ac.za

E-learning support

• Report a problem you experience to the Student Help Desk.

• Email studenthelp@up.ac.za

13 GENERAL
13.1 Announcements
The study guide does not necessarily contain all the information. Important an-
nouncements may be made during recorded sessions and may be published on clickUP.

11
13.2 Calculators
A pocket calculator is essential during tests and examinations.

13.3 ClickUP
All important information will be published on clickUP. It will be assumed that
students are familiar with the contents of all the announcements and material posted
on clickUP.

13.4 Previous tests and examination papers

NO queries with regard to class tests, semester tests and examination papers of
previous years will be answered.

13.5 Emails
The lecturers will not answer emails sent to their personal accounts. Please follow
the procedure indicated in the SWK122 clickUP homepage to communicate with the
lecturer team.

13.6 Writing the examination or the supplementary examination without qual-

ifying
Any student that writes the examination or the supplementary examination without
qualifying (refer to the module requirements in point 6), will be taken to the Disci-
plinary Committee. Refrain from doing so, as it is in violation of EBIT’s regulations
and rules.

13.7 Perusal
• Perusal date

o The perusal date of the examination scripts will be announced on clickUP.

o It is the responsibility of the student to make sure when the perusal takes
place.
o A student forfeits the right for perusal of his/her examination script if
he/she does not attend the perusal on the set date.

• The procedures for the perusal will be communicated via clickUP once the
examination has been completed.

• The perusal is an opportunity for the student to see their examination script
and not a platform for bargaining for more marks to get a distinction/pass the
module/qualify for the supplementary examination.

13.8 Bargaining for marks

Refrain from entering in a discussion for more marks with the lecturers.

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13.9 Formula page
A formula page to be used in Theme 4, is available in the back of the study guide.

13
STUDY COMPONENT
1 USE OF THE STUDY GUIDE
1.1 The module is divided into a number of THEMES. The OBJECTIVES for each
theme provide the student with an overview of the structure of the module.
Each theme is subdivided into LECTURE UNITS with its own LEARNING
OUTCOMES (refer to point 1.4 below). These learning outcomes are essential
to achieve the critical learning outcomes as set out in point 5.
The number of LECTURES and the number of NOTIONAL HOURS as indi-
cated at each lecture unit may differ from the actual time spent on the lecture
unit.
The notional hours include the contact time (formal lectures and tutorials)
and the estimated self study time (preparation for tutorial classes, class tests,
semester tests and examinations).
The indicated notional hours is an average, the actual time to successfully
complete a lecture unit depends on the abilities and circumstances of each
student.

1.2 The material that has to be mastered is indicated under the heading SOURCE
and in the learning outcomes. This material is the minimum required to achieve
the learning outcomes satisfactorily. Unless indicated otherwise, you must com-
prehend and know everything in full.

1.3 The prescribed textbook is used to teach the student how to use a textbook.
This means that the student must become accustomed with the textbook and
be able to do further reading and comprehend on topics not covered in the
module.
This study guide has been compiled referring to the prescribed textbook. Hence
every student must have the textbook.

1.4 The LEARNING OUTCOMES are basic guidelines. It does not mean that
test and examination questions will consist only of theory and the type of
problems spelt out in the learning outcomes. It may be necessary to combine
your knowledge of different lecture units and themes to solve a problem. The
first step however remains to ensure that you have reached the objectives for
each lecture unit.

1.5 Typical problems are listed under SELF STUDY ACTIVITIES.

Partial solutions for the preliminary and fundamental problems are given in the
back of the textbook and may be helpful.
The problems test whether you have reached the learning outcomes of the
lecture units and the objectives of the themes.
You will get the necessary training in the application of your knowledge by
solving the problems. It is of the utmost importance that these problems are

14
 done as soon as possible after the completion of a lecture unit. In this way you
ensure that you do not lapse behind.

2 GENERAL OBJECTIVES
The general objectives of this module are

• to convey the basic principles of mechanics and

• to acquire the basic skills for applying these principles in engineering problems.

The principles are presented to the student by

• explaining the concepts,

• formalizing the concepts in a theory and
• discussing the application of the concepts in examples.

The theory is applied to engineering problems as examples during lectures and tuto-
rials by

• generating mathematical models for specific physical situations,

• solving the mathematical problems that describe the model and
• interpreting the solutions.

Solving problems will assist the student in acquiring the skills to apply the theory.

The mathematical models can only be constructed after the concepts and properties
of forces and moments, which are developed in the theory, have been mastered. It
therefore does not make sense to attempt the exercises without first studying the
relevant theory.

3 PRE-KNOWLEDGE
In this module it is required from the student to apply his/her knowledge of math-
ematics and general science acquired at school, as well as the knowledge of calculus
and vector algebra acquired in the WTW 158 module. It is the student’s own re-
sponsibility to do the necessary revision in good time.

4 ARTICULATION WITH OTHER MODULES IN THE

PROGRAMME
The concept of “force” is of fundamental importance in all branches of engineering.
Therefore all engineering students are required to have a basic knowledge of the
principles of mechanics. This module is fundamental and links up with almost all
other modules presented in the School for Engineering.

15
5 CRITICAL LEARNING OUTCOMES
The following ECSA (Engineering Council of South Africa) graduate attributes are
addressed in the module.

• Graduate attribute 1: Problem solving.

Students acquire the skill to identify, formulate, analyse and solve real world
equilibrium problems.

• Graduate attribute 2: Application of scientific and engineering knowl-

edge.
A systematic, fundamental and theory-based approach is followed to solve two-
and three-dimensional rigid body equilibrium problems. Students learn to draw
free-body diagrams, model forces and moments as mathematical vectors and
apply the principle of equivalent force systems to analyse engineering problems
mathematically.
As application, two-dimensional simple trusses, frames and machines are dis-
cussed, as well as shear force and bending moment diagrams for beams.
Some basic geometric properties of bodies, namely centres of gravity, centroids
and moments of inertia are discussed. Methods to calculate these parameters
for composite regions with the aid of tabulated values are discussed.

• Graduate attribute 9: Independent learning ability.

The development of learning skills such as understanding fundamental concepts,
applying scientific logic and reason and the extensive use of the prescribed
textbook in the study program are emphasised in this module. This facilitates
capacity for independent learning. The student is exposed to a formal structure
of learning, which is also followed in other disciplines.

6 MODULE STRUCTURE
The subject matter for the module is divided into five themes. Themes 1 and 2
describe equilibrium in mathematical terms. Mathematical techniques are used to
solve the problems. Themes 3 and 5 are applications of equilibrium in specific fields.
Theme 4 studies some geometrical properties of regions.

Theme 1: Particle equilibrium

Theme 2: Rigid body equilibrium
Theme 3: Structural analysis
Theme 4: Centroids and second moments of area
Theme 5: Internal forces in structural members

16
THEME 1 PARTICLE EQUILIBRIUM

Number of lectures: 10

Number of notional hours: 34 hours

Objectives

In this theme we model equilibrium of a particle by applying Newton’s first and third
laws.

The physical concept “force” is introduced and the mathematical description (mod-
elling) of force vectors is discussed in detail. The word “vector” originates from the
Latin word “vehere” which means “to carry”. Force vectors were first used by the
18th century astronomers who studied planet rotation around the sun.

Regarding forces mathematically as vectors, give us a means to solve problems math-

ematically.

The force concept is modelled mathematically with a geometrical definition. When

problems are solved with the geometrical interpretation of vectors, the solving tech-
niques can become complex. To simplify the solving process, we interpret force
vectors algebraically (as cartesian vectors). It is important that the student is com-
fortable with both the geometrical and algebraic interpretations of vectors.

PRE-KNOWLEDGE AND SELF STUDY

Vectors and their properties as discussed in WTW 158. The relevant material is
found in CALCULUS, Stewart, Sections 12.1 – 12.3.

LECTURE UNIT 1.1 GENERAL PRINCIPLES

Source: Textbook Sections 1.1 to 1.5 (pp 2 – 15).

Class notes and any undergraduate linear algebra textbook.

Learning outcomes

After completion of this unit you must

1. be familiar with
(a) basic and derived quantities and idealizations in mechanics,
(b) Newton’s laws of motion,
(c) the international system of units (SI units),
(d) standard procedures used in numerical calculations,
(e) general procedures for analysing problems,
2. be able to

17
 (a) round numbers to a specified number of significant digits or decimal digits,
(b) solve system of linear equations,
(c) determine whether a system of linear equations has
(i) a unique solution,
(ii) infinitely many solutions (a parametric solution) or
(iii) no solution,
(d) use the sine rule and cosine rule to determine unknown lengths and angles
in triangles.

Remarks

1. Use g = 9.81 m/s2 , unless stated otherwise.

2. The number of significant digits and the number of decimal digits in a number
are two different concepts.

3. When doing any calculation, except for angles, use at least four (4) significant
digits in intermediate steps and round the final answer to four (4) significant
digits, unless stated otherwise.

4. In the case of angles, used at least (1) decimal digits in intermediate steps in
calculations and round the final answer to one (1) decimal digit, unless stated
otherwise.

5. Textbook answers and lecture preparation snippet answers are not necessarily
given to the accuracies as described in points 3 and 4.

6. Add your units in each step of your calculations.

7. Write coherent and clearly and use “ = ” signs in your calculations.

Self study activities

Textbook p 15: Problems 1.2, 1.5, 1.8, 1.12, 1.13, 1.17, 1.18, 1.21.

LECTURE UNIT 1.2 GEOMETRIC VECTOR OPERATIONS

Source: Textbook Sections 2.1, 2.2 and 2.3 (pp 16 – 32).

Learning outcomes

After completion of this unit you must

1. know the meaning of the following terminology:

(a) scalar,

18
 (b) vector,
(c) magnitude of a vector,
(d) line of action of a force vector,
(e) sense and direction of a vector,
(f) components of a vector,
(g) collinear vectors,
(h) resultant of a system of vectors,

2. be able to

(a) to represent force vectors graphically,

(b) describe a force mathematically as a vector,
(c) add and subtract vectors with the parallelogram rule,
(d) multiply a vector with a scalar using the geometrical definition,
(e) resolve a vector in given components (the components do not necessarily
have to be perpendicular on each other),

3. solve problems using the above mentioned techniques.

Remarks

1. The triangle rule for addition of two vectors follows from the parallelogram rule.

• The triangle rule is therefore merely a mathematical tool in determining

the magnitude and direction of vectors.
• The lines that describe the triangle do not necessarily coincide with the
lines of action of the forces.
• The points of application of the forces can not be deduced from the infor-
mation of the triangle.

2. “Division of vectors” is undefined.

3. Resolving a vector in given components is not a unique process. Each vector

can be resolved in infinitely many ways, depending on the choice of components.

Self study activities

1. Textbook p 27: Preliminary problems P2.1 – P2.2.

2. Textbook p 28: Fundamental problems F2.1 – F2.6.

3. Textbook pp 29 – 32: Problems 2.16, 2.22.

19
 4. Textbook p 29: Refer to sketch Probs 2-4/5. Find F and θ such that the
resultant of the given forces on the hook has magnitude 800 N . The angle of
the resultant measures 120◦ in the anticlockwise direction from the positive
x−axis.

5. Textbook p 29: Refer to sketch Probs 2-6/7. Set FB = 2 kN and θ = 45◦ and
calculate the components of FB along the u−axis and the y−axis.

6. Textbook p 39: Refer to sketch F2-10. The resultant of the three forces has
magnitude 600 N . Calculate the minimum value and the maximum value that
F can attain, as well as the corresponding angle θ in each case. Hint: First
calculate the resultant of the two known forces.

7. Textbook p 278: Refer to sketch Fig 6-8 p 278. Calculate the components of
the 500 N force along the members AB and BC.

LECTURE UNIT 1.3 ALGEBRAIC VECTOR ADDITION WITH

RECTANGULAR COMPONENTS IN A PLANE

Source: Textbook Section 2.4 (pp 33 – 43).

Learning outcomes

After completion of this unit you must

1. be familiar with

(a) cartesian vector forms of a vector,

(b) the symbols i and j (denoted by i and j in the textbook),

2. be able to

(a) express vectors in cartesian vector form,

(b) find the decomposition of a vector along specified rectangular axes (the
components are also called perpendicular components of the vector and
depend on the choice of the system of axes),
(c) find the resultant of a system of coplanar forces acting on a common point
using rectangular / perpendicular components algebraically (i.e. with the
aid of cartesian vector notation),
(d) understand and be able to interpret the meaning of the signs (“ + ” or
“ − ”) of the scalars in a cartesian vector.

Self study activities

1. Textbook p 39: Fundamental problems F2.7 – F2.12.

2. Textbook pp 40 – 43: Problems 2.39, 2.40, 2.55.

20
 3. Textbook p 43: Refer to the sketch Probs. 2-55/56 and answer the following
questions. Describe all vectors by (i) specifying the magnitude of the vector and
(ii) expressing the direction as the angle that the vector has with the positive
x−axis, measured anticlockwise.

(a) Determine the vector F 23 ≡ F 2 + F 3 .

(b) Let φ = 30◦ . Determine F 1 and R ≡ F 1 +F 2 +F 3 so that R is a minimum.
(c) Determine F 1 , R ≡ F 1 + F 2 + F 3 and φ so that R is a minimum.
(d) Determine F 1 , R ≡ F 1 + F 2 + F 3 and φ so that R lies along the u−axis
and F1 is a minimum.
(e) Determine the minimum value for F1 and the corresponding angle φ so
that the resultant force that acts on the hook, has magnitude 500 N .
(f) Determine the maximum value for F1 and the corresponding angle φ so
that the resultant force that acts on the hook, has magnitude 500 N .
(g) Determine the minimum value for F1 and the corresponding angle φ so
that the resultant force that acts on the hook, has magnitude 300 N .
(h) Determine the maximum value for F1 and the corresponding angle φ so
that the resultant force that acts on the hook, has magnitude 300 N .

LECTURE UNIT 1.4 PARTICLE EQUILIBRIUM, FREE-BODY

DIAGRAMS AND MODELLING ASSUMPTIONS

Source: Textbook Sections 3.1 and 3.2 (pp 86 – 90).

Learning outcomes

After completion of this unit you must

1. know and be able to formulate the vector equation that ensures equilibrium of
a given force system on a particle,

2. know, be able to formulate and apply the modelling assumptions for springs,
cables and pulley systems in problems,

3. know and understand what is meant by a free-body diagram of a particle,

4. be able to draw the free-body diagram of a particle.

Self study activities

1. Textbook p 96: Preliminary problem P3.1.

2. Draw the free-body diagrams of the indicates particles in the following prob-
lems.

21
 (a) Textbook p 97: Fundamental problems F3.1 (A), F3.2 (B), F3.3 (B),
F3.4(a) (block), F3.5 (B & E), F3.6 (B & C).
(b) Textbook pp 98 – 105: Problems 3.3 (O), 3.9 (B & C), 3.10(b) (B), 3.14
(A & B), 3.16 (A), 3.24(c) (sphere A and pulley B), 3.28 (B & E), 3.32
(A, B, C & D).

Remarks and hints on the self study activities

(a) The dimensions of the block are not given, hence consider the block as a particle.
(b) Note that the points A, B, C and D are coplanar.
(c) The sphere and pulley are modelled as particles.

LECTURE UNIT 1.5 PARTICLE EQUILIBRIUM IN TWO DIMENSIONS

Source: Textbook Section 3.3 (pp 91 – 105).

Learning outcomes

After completion of this unit you must

1. know what is meant by concurrent and coplanar force systems,

2. be able to derive the applicable algebraic (scalar) equilibrium equations for
coplanar concurrent force systems from the general vector formulation,
3. be able to apply the algebraic equilibrium equations on coplanar concurrent
force systems to analyse equilibrium problems.

Remarks

1. The abbreviation FBD for “free-body diagram” may be used in tests and ex-
aminations.
2. Make sure that you always draw a clear and separate FBD as a first step. This
is non-negotiable since
• the directions of the vectors that you use are indicated in the FBD,
• the vectors are labelled in the FBD, and
• the reader can follow your steps.
The marker may refuse to mark your questions if your answer is not
accompanied by a clear separate FBD.
3. When you have chosen directions for your vectors in a FBD, do not change
them on the FBD in due course. Your equilibrium equations are set due
to these directions and if you change them, your equilibrium equations must be
adapted as well. This will result in an endless circle of changes that confuses
everybody and your chances of making errors are high!!!

22
 4. Indicate yourP steps that you
P follow by stating the equilibrium equations that
you use (i.e. Fx = 0 or Fy = 0). This ensures a clear and organised way
to represent the relationships in your available information.

Self study activities

1. Textbook p 96: Preliminary problem P3.2.

2. Refer to the FBD’s drawn in the problems of Lecture Unit 1.4.

(a) Textbook p 97: Fundamental problems F3.1 (A), F3.2 (B), F3.3 (B),
F3.4(a) (block), F3.5 (B & E), F3.6 (B & C).
(b) Textbook pp 98 – 105: Problems 3.3 (O), 3.9 (B & C), 3.10(b) (B), 3.14
(A & B), 3.16 (A), 3.24(c) (sphere A and pulley B), 3.28 (B & E), 3.32
(A, B, C & D).

LECTURE UNIT 1.6 CARTESIAN VECTORS IN THREE DIMENSIONS

Source: Textbook Sections 2.5 and 2.6 (pp 44 – 55).

Learning outcomes

After completion of this unit you must

1. be familiar with

(a) the symbols i, j and k (denoted by i, j and k in the textbook),

(b) right-handed rectangular coordinate systems and
(c) unit vectors,

2. know that for each non-zero vector, unit vectors parallel to the given vector
exist, with the sense of the unit vectors the same or opposite as the sense of
the original vector,

3. understand the meaning of the signs (“ + ” or “ − ”) of the scalars in a vector

decomposition,

4. be able to find the three orthogonal (perpendicular / rectangular) components

of a vector in three dimensions (which relates to the cartesian form of the vector
and the system of axes that is used),

5. know what is meant by direction angles and direction cosines of a vector and
know how to calculate these values,

6. be able to formulate and use the relationship between the

(a) direction cosines of a vector,

(b) the rectangular components and the direction cosines of a vector,

23
 7. be able to express a vector in terms of its magnitude and a unit vector parallel
to the given vector,

8. to find the resultant of a system of vectors using the cartesian vector represen-
tations of the vectors,

9. be able to use and apply the above mentioned techniques in problems.

Self study activities

1. Textbook p 50: Preliminary problems P2.3 – P2.5.

2. Textbook p 51: Fundamental problems F2.13 – F2.18.

3. Textbook pp 52 – 55: Problems 2.61, 2.66, 2.68, 2,69, 2.80, 2.81.

LECTURE UNIT 1.7 POSITION VECTORS AND VECTORS

DIRECTED ALONG A LINE

Source: Textbook Sections 2.7 and 2.8 (pp 56 – 68).

Learning outcomes

After completion of this unit you must

1. (a) be familiar with the concept of position vectors,

(b) be familiar with the concepts of a fixed vector and a sliding vector,
(c) understand why force vectors are considered as sliding vectors and position
vectors as position vectors in statics,

2. be able to

(a) calculate position vectors with know starting and end points,
(b) express any vector parallel to a second vector in terms of the second vector,

3. be able to solve problems using relevant information on position vectors.

Self study activities

1. Textbook p 63: Preliminary problems P2.6 – P2.7.

2. Textbook p 64: Fundamental problems F2.19 – F2.24.

3. Textbook pp 65 – 68: Problems 2.86, 2.89, 2.96, 2.97.

24
LECTURE UNIT 1.8 PARTICLE EQUILIBRIUM IN THREE
DIMENSIONS

Source: Textbook Section 3.4 (pp 106 – 116).

Learning outcomes

After completion of this unit you must be able to derive the applicable algebraic
equilibrium equations and apply them to analyse equilibrium problems for a general
force system on a particle.

Self study activities

1. Draw the free-body diagrams of the indicated particles in the following prob-
lems.

(a) Textbook p 111: Fundamental problems F3.7 (the particle at the origin),
F3.8 – F3.11 (A).
(b) Textbook pp 112 – 116: Problems 3.51 (A), 3.53 (O), 3.58 (A).

2. Problems 3.51, 3.55, 3.67.

LECTURE UNIT 1.9 DOT PRODUCT AND APPLICATIONS

Source: Textbook Section 2.9 (pp 69 – 80).

Learning outcomes

After completion of this unit you must

1. be able to formulate the geometrical and algebraic definitions of the dot product,

2. know that the geometrical and algebraic definitions of the dot product are
equivalent (see Remark 1),

3. know and be able to formulate properties of the dot product (see Remark 2),

4. know the meaning of the projection of a vector on a given line or vector,

5. be able to find the components of any vector parallel and perpendicular to a

given line or vector,

6. be able to use the dot product and its properties in solving problems.

Remarks

1. Definition A and Definition B are equivalent if you can prove that

(a) the formulation of Definition B follows from Definition A and that

25
 (b) the formulation of Definition A follows from Definition B.

2. The properties of the dot product can be proven by taking any one of the two
formulations of the dot product as definition.

Self study activities

1. Textbook p 74: Preliminary problems P2.7 – P2.8.

2. Textbook p 75: Fundamental problems F2.25 – F2.31.

3. Textbook pp 76 – 80: Problems 2.111, 2,113, 2.130(d) , 2.134(e) .

Remarks and hints on the self study activities

(d) Express the components in cartesian vector form. Explain why F 1 and F 2 do not have
projections on the x−, y− and z−axes.
(e) Use the geometric definition of the dot product for finding θ.

26
THEME 2 RIGID BODY EQUILIBRIUM

Number of lectures: 13

Number of notional hours: 44 hours

Objectives

Theme 1 deals with particle equilibrium, where the force system acting on the particle
is concurrent.

In Theme 2 we generalise the concept of equilibrium of concurrent force systems

to equilibrium of rigid bodies, where the dimensions of the body is of importance.
Two new concepts, moments of forces and couple moments, are introduced. The
equilibrium of a rigid body is described in terms of the forces acting on the body,
together with the moments that develop due to the force system.

The underlying mathematics for rigid body equilibrium and reducing a force system
to its simplest form will be discussed.

PRE-KNOWLEDGE AND SELF STUDY

The cross product between vectors and the properties of the cross product as discussed
in WTW 158. The relevant material is found in CALCULUS, Stewart, Section 12.4.

LECTURE UNIT 2.1 THE MOMENT OF A FORCE

Source: Textbook Sections 4.1, 4.2, 4.3 and 4.4 (pp 120 – 144).

Learning outcomes

After completion of this unit you must

1. be able to define and determine

(a) the cross product between two vectors and

(b) the moment of a force about a point P (or the related line perpendicular
to the plane containing the force and the point P )

geometrically and algebraically,

2. know and be able to apply the properties of the cross product,

3. be familiar with the relationship between the mathematical description and the
physical properties of the moment of a force about a point (or line as described
in learning outcome 1(b)),

4. be able to interpret the moment of a force in two dimensions

27
 (a) in terms of clockwise (negative) and anticlockwise (positive) moments,
(b) as a special case of the cartesian vector moment in three dimensions,

5. be able to determine the resultant moment of a force system about a point (or
line as described in in learning outcome 1(b)),

6. be able to prove that the moment of a force about a point (or line as in learning
outcome 1(b)) does not depend on which point on the line of action of the force
is chosen in calculating the moment,

7. be able to formulate, prove and apply Varignon’s theorem (principle of mo-

ments).

Remarks

1. We use throughout the sign convention for clockwise and anticlockwise moments
in a plane as discussed in the textbook.

2. Some of the answers in the back of the textbook do not comply with the sign
convention above. As an example, the answer in 4.18 (a) is −73.9 N m. The
textbook however states the answer as 73.9 N m (but does indicate the clockwise
direction as positive). We will not use this method.

3. Note that the formula MO = Fx y − Fy x on page 132 is not a general formula;

but only holds for depiction in Fig. 4 − 17.

4. In numerous three-dimensional equilibrium problems, the algebraic formulation

of the moment of a force about a point may simplify the process of finding
moments about specific lines (axes). The reason for this statement follows:
In the geometrical definition, perpendicular distances to the lines of action of
the forces may be difficult to find. Furthermore, the rotational directions must
also be determined.
The algebraic formulation automatically takes care of rotational directions and
it is not necessary to know the perpendicular distances to the lines of action of
the forces.

5. The algebraic definition can also be applied for determining the moments of two-
dimensional forces and comes in handy for coplanar force systems, especially
when the geometry of the problem becomes tedious to handle.

Self study activities

1. Textbook 135: Preliminary problems P4.1 – P4.2.

2. Textbook pp 136 – 137: Fundamental problems F4.1 – F4.12.

3. Textbook pp 138 – 144: Problems 4.12, 4.14, 4.18, 4.19, 4.44, 4.45, 4.48.

28
 4. Textbook p 140: Refer to sketch Probs 4-22/23 with F = 100 N . Find the
maximum and minimum moment of F about O using the following two ap-
proaches.

• Method 1: Use calculus.

(a) Express the moment M of F about O in terms of θ. (The expression
will be of the form M = a cos θ + b sin θ.)
(b) Rewrite M in the form M = c sin(α + θ) and calculate the numerical
values for c and α. You can now find the maximum and minimum
value(s) of M from your knowledge of the sine graph.
(c) Find the maximum and minimum value(s) of M using differentiation
(first and second order derivatives).
• Method 2: Use geometry.
(a) Use the fact that the minimum moment is 0. When does this happen?
(b) M = F d (with d the perpendicular distance from A to the line of
action of F ) denotes the magnitude of the moment of F about A.
Consider some arbitrary values for θ and interpret graphically the
change in d. Find the situation for which the moment is a maximum
and determine the corresponding d. Calculate the maximum moment
and the corresponding angle.

LECTURE UNIT 2.2 THE MOMENT OF A FORCE ABOUT AN AXIS

OR LINE

Source: Textbook Section 4.5 (pp 145 – 153).

Revision: Textbook Section 2.9 (pp 69 – 80).

Learning outcomes

After completion of this unit you must be able to

1. define the moment of a force about an axis or line,

2. determine the moment of a force about an axis or line by using the geometrical
and the algebraic formulation.

Self study activities

1. Textbook p 150: Preliminary problems P4.3 – P4.4.

2. Textbook p 151: Fundamental problems F4.13 – F4.18.

3. Textbook pp 152 – 153: Problems 4.54(a) , 4.55, 4.63, 4.65.

29
Remarks and hints on the self study activities

(a) Follow the outline below, and answer the extra questions.

(i) Find the cartesian vector moment M A of F about A.

(ii) Find the cartesian vector moment M C of F about C.
(iii) Find the cartesian vector moment M AC of F about AC using (i).
(iv) Find the cartesian vector moment M AC of F about AC using (ii).
(v) What conclusion can you draw from your answers in (iii) and (iv)? Con-
vince yourself that this is in line with the theory.
(vi) What is the relationship between
(a) M AC and M A and (b) M AC and M C ?
(vii) Determine the moments of F about the x−, y− and z−axes. Can you use
any of the calculations in the previous questions to answer this question?
Explain why you say so.

LECTURE UNIT 2.3 THE MOMENT OF A COUPLE

Source: Textbook Section 4.6 (pp 154 – 165).

Learning outcomes

After completion of this unit you

1. must be able to define

(a) a couple,
(b) the moment of a couple (algebraically and in terms of geometrical prop-
erties) and
(c) equivalent couples,

2. must be able to

(a) calculate the moment of a couple using the scalar definition (geometrical
properties) and the algebraic definition,
(b) determine whether given couples systems are equivalent,
(c) calculate the resultant of a system of couples,

3. must be able prove that a couple is a free vector (when applied on a rigid body
in equilibrium).

Self study activities

1. Textbook p 160: Fundamental problems F4.19 – F4.24.

30
 2. Textbook pp 161 – 165: Problems 4.68, 4.73, 4.78, 4.89, 4.90, 4.96(b) .

Remarks and hints on the self study activities

(b) Where does the resultant couple moment act?

LECTURE UNIT 2.4 EQUIVALENT FORCE AND COUPLE SYSTEMS

Source: Textbook Section 4.7 (pp 166 – 171).

Learning outcomes

After completion of this unit you must

1. know that

(a) systems of equivalent forces and couples have the same external effect on
a rigid body in equilibrium,
(b) the external effects on a rigid body in equilibrium tend to cause transla-
tional and rotational motion,
(c) reaction forces and reaction couples at the supports form part of the system
of external forces on a body in equilibrium,

2. be able to replace a force and couple system with an equivalent force and couple
system.

Self study activities

1. Textbook p 172: Preliminary problem P4.5.

2. Textbook p 173: Fundamental problems F4.25 – F4.30.

3. Textbook pp 174 – 176: Problems 4.101(c) , 4.105, 4.109.

Remarks and hints on the self study activities

(c) (i) Answer the question as formulated.

(ii) Replace the force and couple system with an equivalent force and couple
system acting at B.
(iii) Depict your two equivalent systems on two separate sketches of the frame.
(iv) What are the similarities and differences in your two answers?

31
LECTURE UNIT 2.5 EQUILIBRIUM IN TWO DIMENSIONS

Source: Textbook Sections 5.1, 5.2 and 5.3 (pp 206 – 229).

Learning outcomes

After completion of this unit you must be able to

1. formulate the equivalent sets of equilibrium equations in two dimensions,

2. draw free-body diagrams for bodies or parts of bodies in equilibrium,

3. identify support reactions on bodies or parts of bodies,

4. interpret internal forces and moments as external reactions where applicable,

5. model the weight of a body on a FBD (to be discussed in detail in Theme 4),

6. solve equilibrium problems in two dimensions.

Remarks

1. Study Table 5.1 pp 210 – 211 for typical support reactions. Weightless links
behave as two-force members and will be discussed in Lecture Unit 2.6.

2. Follow the steps set out below when solving equilibrium problems:

(a) Identify a part of the body which has to be studied and draw a FBD of
this part. The FBD consists of
(i) an outline of the part of the body,
(ii) all relevant information, i.e. forces, moments, distances, slopes and
angles,
(iii) all the unknowns; label these unknowns with typical symbols for ex-
ample F , Ax , Ay , M etc.
(b) Set up the applicable system of equilibrium equations according to the
directions for the forces and moments on your FBD.
(c) Number the equations mentioned above and solve for the unknowns. Read
again the remarks of Lecture unit 1.5.
(d) Interpret your answers. For instance, if you find the answers F = 100 N
and P = −5 kN , the direction of the force F in your FBD is chosen correct
and the direction of the force P is chosen in the wrong direction.

3. In Example 5.7 p 224, the so-called triangular distributed loading (the triangle
with the downward arrows), is replaced by a single force of 60 N acting 1 m
from point A. The principle behind this replacement is discussed in Lecture
Unit 4.2.

32
Self study activities

1. Textbook p 219: Sketch the free-body diagrams as stipulated AND write down
the system of equilibrium equations in each case. You do not have to solve the
systems of equations.
Problems 5.1 (a) & (b), 5.2 (b) & (c), 5.4 (a) & (c), 5.5 (c) & (d), 5.7 (a).

LECTURE UNIT 2.6 TWO-FORCE AND THREE-FORCE MEMBERS

Source: Textbook Section 5.4 (pp 230 – 244).

Learning outcomes

After completion of this unit you must

1. know the properties of two-force and three-force members,

2. be able to prove that the properties of two-force and three-force members re-
ferred to above, are valid,

3. be able to apply the properties of two-force and three-force members in equi-

librium problems.

Self study activities

In each of the listed problems, identify the two-force members and three-force mem-
bers and use this information in determining the required reactions.

1. Textbook p 232: Preliminary problem P5.1 (a), (b), (d) & (f).

2. Textbook pp 233: Fundamental problems F5.1 – F5.6.

3. Textbook p 234 – 244: Problems 5.11, 5.13, 5.16, 5.38, 5.45, 5.54.

LECTURE UNIT 2.7 EQUILIBRIUM IN THREE DIMENSIONS

Source: Textbook Sections 5.5, 5.6 and 5.7 (pp 245 – 267).

Learning outcomes

After completion of this unit you must be able to

1. draw free-body diagrams for bodies or parts of bodies in equilibrium,

2. formulate equilibrium equations in three dimensions,

3. identify support reactions on bodies or parts of bodies,

33
 4. solve equilibrium problems in three dimensions,

5. identify if a problem is (i) statically determinate, (ii) statically indeterminate

or (iii) improperly constrained.

Remarks

1. Table 5.2 pp 246 – 247 show typical support reactions. Study the table.

2. Example 5.14 p 249 discusses examples of bodies for which the supports are
properly aligned.

Self study activities

1. Textbook p 260: Preliminary problems P5.2 – P5.3.

2. Textbook p 261: Fundamental problems F5.7 – F5.12.

3. Textbook pp 262 – 267: Problems 5.68, 5,70, 5.80, 5.84, 5.85.

LECTURE UNIT 2.8 REDUCTION OF A FORCE AND COUPLE

SYSTEM TO ITS SIMPLEST FORM

Source: Textbook Section 4.8 (pp 177 – 189).

Learning outcomes

After completion of this unit you must be able to

1. define a wrench,

2. formulate conditions under which a force and couple system can be reduced to
a

(a) single force,

(b) wrench,

3. motivate that it is mathematically always possible to replace a force and couple

system in two dimensions with a single force,

4. reduce a given force and couple system to its simplest form, which is either a
single force or a wrench.

Self study activities

1. Textbook p 184: Preliminary problems P4.6 – P4.7.

34
2. Textbook p 185: Fundamental problems F4.31 – F4.36.

3. Textbook pp 186 – 189: Problems 4.126, 4.131.

4. Find the simplest equivalent force system to the given force system on the pipe
assembly. Will it be possible to apply this simplest force system on the pipe
assembly? Motivate your answer in full.
z

N
0i
12
D 0.3 m

−
0.3 m

N
0i
0.5 m

12
0.8 m B 0.7 m
A y
C
60k N

m
6
0.

R
P Q
−60k N

x P Q = QR = 0.25 m

−40i N

35
THEME 3 STRUCTURAL ANALYSIS

Number of lectures: 5

Number of notional hours: 17 hours

Objectives

In Theme 3 we apply the fundamentals of rigid bodies equilibrium as discussed in

Themes 1 and 2, to simple trusses, frames and machines. The underlying theory,
modelling assumptions and methods to determine the loads that the members and
pin-connections carry, are discussed.

We only consider two-dimensional problems. The three-dimensional case is a general-

ization of the two-dimensional case – the applied principles in analysing the structures
are the same.

LECTURE UNIT 3.1 SIMPLE TRUSSES

Source: Textbook Sections 6.1 – 6.4 (pp 272 – 290).

Learning outcomes

After completion of this unit you must

1. know and be able to list the modelling assumptions for simple trusses,

2. be able to apply the method of joints in analysing a truss,

3. be able to apply the method of sections in analysing a truss,

4. be able to conclude whether a member is in tension or in compression,

5. define and identify zero-force members,

6. know how to include the weight of members such that the modelling assump-
tions stay valid.

Remark

Both the method of joints and the method of sections are direct applications of rigid
body equilibrium.

Self study activities

1. Textbook p 285: Preliminary problems P6.1 – P6.2.

2. Textbook p 286: Fundamental problems F6.1 – F6.6.

36
 3. Textbook pp 287 – 290: Use the method of joints for the following problems.
Problems 6.5, 6.9, 6.11, 6.17(a) .
4. Textbook p 297: Fundamental problems F6.7 – F6.12.
5. Textbook pp 298 – 300: Use the method of sections for the following problems.
Problems 6.31(b) and 6.35.

Remarks and hints on the self study activities

(a) Follow the outline below to solve the problem.

(i) Use the fact that due to symmetric vertical loading, the problem is sym-
metric around member JD. (If the external loading changes, symmetry is
not guaranteed.)
(ii) Determine all the zero-force members.
(iii) Apply the method of joints repeatedly to determine the forces in the mem-
bers.
(b) You will need two section cuts to determine the asked information.

LECTURE UNIT 3.2 FRAMES AND MACHINES

Source: Textbook Section 6.6 (pp 305 – 336).

Learning outcomes

After completion of this unit you must

1. be able to distinguish between trusses, frames and machines,

2. know and be able to list the modelling assumptions for frames and a machines,
3. know that in frames and machines at least one member is a multi-force member,
4. be able to determine whether a two-force member in a frame or machine is in
tension or in compression.

Remark

It is meaningless to refer to a multi-force member to be in “compression” or “tension”.

Self study activities

1. Textbook p 321: Preliminary problem P6.3 (a), (c), (e) & (f).
2. Textbook pp 322 – 324: Fundamental problems: F6.13 – F6.24.
3. Textbook pp 325 – 336: Problems 6.63, 6.68, 6,83, 6.99, 6.117.

37
THEME 4 CENTROIDS AND SECOND MOMENTS OF AREA

Number of lectures: 5

Number of notional hours: 17 hours

Objectives

Theme 4 deals with centroids and second moments of area of certain types of bodies.
These properties are of geometrical nature and they are derived from the general
concepts centers of gravity and moments of inertia respectively.

The applications of centroids are numerous and our emphasis is on centroids of two-
dimensional areas and curves. This however does not exclude centers of gravity and
other related properties.

Moments of inertia are quantities that measure the ability of structures to resist
bending. There are different types of moments of inertia and we will discuss only
moments of inertia for areas, also called second moments of area.

Calculating centroids and second moments of area with integration is not included
in the syllabus.

LECTURE UNIT 4.1 CENTROIDS

Source: Textbook Section 9.2 (pp 488 – 501) and class notes.

Learning outcome

After completion of this unit you must be able calculate centroids of curves in two
and three dimensions, as well as centroids of areas in a plane.

Remark

The table in the back of the study guide will be provided during tests and examina-
tions.

Self study activities

1. Textbook p 493: Fundamental problems F9.7 – F9.12.

2. Textbook pp 494 – 501: Problems 9.56, 9.58, 9.67, 9.69.

3. Locate the centroid of the wire.

38
 y

20 mm
x
m
m
15

4. In the frame below, the mass per unit length for member BD is half of the mass
per unit length for members AC and CE, which are equal. Locate the position
(x, y) of the center of gravity. Member AC has length 13 m and member CE
has length 8 m. Point B lies half way between points A and C and point D lies
half way between C and E.
y

C D E
b b

B b

A
x

5. Calculate the centroid of the composite area.

z 3m
3m

6m
3m
y

10 m

6. Calculate the centroid of the composite region consisting of the triangle (T )

and the wedge (W ).

39
 y (cm)

1.5
T 1
x (cm)
50◦
W

LECTURE UNIT 4.2 SIMPLE DISTRIBUTED LOADINGS

Source: Textbook Section 4.9 (pp 190 – 200).

Learning outcomes

After completion of this unit you must be able to replace a simple distributed loading
with an equivalent system of concentrated loads.

Remark

Example 4.21, Fundamental problem F4.42 and Exercises 4.157 – 4.162 are not in
the scope of this module.

Self study activities

1. Textbook p 195: Fundamental problems F4.37 – F4.41.

2. Textbook pp 196 – 199: Problems 4.140, 4.151, 4.153, 4.156.

LECTURE UNIT 4.3 SECOND MOMENTS OF AREA

Source: Textbook Sections 10.1, 10.2 and 10.4 (pp 528 – 531 & pp 540 – 547) and
class notes.

Learning outcomes

After completion of this unit you must

1. know what the moments of inertia Ix and Iy and the polar moment of inertia
JO mean,
2. know that Ix + Iy = JO , where O is the origin of a (rectangular) xy−system of
axes,
3. be able to formulate and apply the parallel-axis theorem for an area (Steiner’s
theorem),

40
 4. calculate moments of inertia (second moments of area) of composite areas.

Remark

Examples 10.1 – 10.3, Fundamental problems F10.1 – F10.4 and Exercises 10.1 –
10.24 are not in the scope of this module.

Self study activities

1. Textbook p 543: Fundamental problems F10.5 – F10.8.

2. Textbook pp 544 – 547: Problems 10.25, 10.31, 10.46, 10.47.

3. Calculate the second moments of area Ix and Iy of the composite region con-
sisting of the rectangular triangle (T ) and the wedge (W ) with radius r. The
wedge is centered symmetrically about the y−axis and spans an angle of 30◦ .
Express your answer in terms of r.

T
x

41
THEME 5 INTERNAL FORCES IN STRUCTURAL MEMBERS

Number of lectures: 8

Number of notional hours: 27 hours

Objective

Internal forces and moments at any point in a rigid body depends on the external
loading on the body.

In this theme we study the relationships between external and internal loadings,
focussing on beams. The principles can be generalised to other situations as well.

Our objective is to determine the shear force and bending moment at each cross
sectional area of a beam.

PRE-KNOWLEDGE AND SELF STUDY

1. Piecewise continuous functions and their graphs.

2. The relationship between a function f , its derivative f 0 and integral

R
f.

3. Drawing the graph of f 0 when f is known.

R
4. Drawing the graph of f when f is known.

Source: Any calculus textbook.

LECTURE UNIT 5.1 INTERNAL LOADING AT A CROSS SECTION

OF A BEAM

Source: Textbook Section 7.1 (pp 342 – 360).

Learning outcomes

After completion of this unit you must be able to

1. describe the shear force, normal force and bending moment at a cross section
of a beam,

2. calculate the internal loading that develops in a given cross section of a beam
due to the external loading applied on the beam.

Self study activities

1. Textbook p 351: Preliminary problem P7.1.

42
 2. Textbook p 352: Fundamental problems F7.1 – F7.6.

3. Textbook pp 353 – 360: Problems 7.9, 7.19, 7.23, 7.27, 7.30.

LECTURE UNIT 5.2 SHEAR FORCE AND BENDING MOMENT

FUNCTIONS AND DIAGRAMS

Source: Textbook Section 7.2 (pp 361 – 369).

Learning outcomes

After completion of this unit you must

1. know the sign conventions for positive shear and positive bending moment as
defined in the textbook,

2. be able to determine the shear force function (V ) and bending moment function
(M ) for a loaded beam in an arbitrary cross section, using the method of
sections,

3. be able to draw the shear force and bending moment diagrams for a given
loaded beam.

Remarks

1. The shear force and bending moment functions are expressed in terms of a
variable (e.g. x) that measures the distance from a fixed reference point to
the cross section.

2. We use throughout the sign convention as discussed in the textbook.

Self study activities

1. Textbook p 365: Fundamental problems F7.7 – F7.12.

2. Textbook pp 366– 369: Problems 7.47, 7.56, 7.57. 7.58.

LECTURE UNIT 5.3 RELATIONSHIP BETWEEN THE DISTRIBUTED

LOAD FUNCTION , SHEAR FORCE FUNCTION
AND BENDING MOMENT FUNCTION

Source: Textbook Section 7.3 (pp 370 – 380).

Learning outcomes

After completion of this unit you must

43
 1. know the relationship between the distributed load function (w), shear force
function (V ) and bending moment function (M ).

2. be able to draw the shear force and bending moment diagrams using these
relationships.

Remarks

1. The derivation of the relationships between w, V and M is not required for

tests and examinations.

2. It is important to show all relevant information on the shear force and bending
moment diagrams. This includes

• the correct form for the diagrams (convex and concave),

• points of intersections of the graphs with the axes,
• local maxima and minima values of the graphs and where these values are
attained,
• inflection points and where the concavity changes,
• discontinuities in the V −function and M −function, which are represented
by“jumps” and “drops” (straight vertical lines) in the graphs.

Self study activities

1. Textbook p 376:: Fundamental problems F7.13 – F7.18.

2. Textbook pp 377 – 380: Problems 7.71, 7.82, 7.86.

44
ANSWERS TO THE EXTRA PROBLEMS IN THE STUDY GUIDE

The answers of the extra problems as formulated in the self study activities, follow.

LECTURE UNIT 1.2

4. F = 916.6 N and θ = 72.5◦ .

5. Fu = 1.633 kN and Fy = 0.5977 kN .

6. Fmin = 36.90 N with corresponding angle θ = 347.3◦ .

Fmax = 1 163 N with corresponding angle θ = 167.3◦ .

7. FAB = 500.0 N and FBC = 707.1 N .

LECTURE UNIT 1.3

√
3 (a) F23 = 60 41 kN = 384.2 kN and θF23 = 321.3◦ .
(b) R = 379.8 N and θR = 330.0◦ , F1 = 57.84 N and θF1 = 60.0◦ .
(c) The minimum resultant is R = 0 N , hence F 1 = −F 23 and F1 = 384.2 kN .
The corresponding angle is θF1 = φ = 141.3◦ .
(d) R = 139.8 N and θR = 30.0◦ , F1 = 357.8 N and φ = 330.0◦ .
(e) F1 = (500 − 384.2) N = 115.8 N and θF1 = θF23 = φ = 321.3◦ .
(f) F1 = (500 + 384.2) N = 884.2 N and θF1 = θF23 − 180◦ = φ = 141.3◦ .
(g) F1 = (384.2 − 300) N = 84.2 N and θF1 = θF23 − 180◦ = φ = 141.3◦ .
(h) F1 = (384.2 + 300) N = 684.2 N and θF1 = θF23 − 180◦ = φ = 141.3◦ .

LECTURE UNIT 1.9

2 (d) Hibbeler edition 14 p 79. Problem 2.128. Point A does not lie on any one of
the three axes.

LECTURE UNIT 2.1

4. Hibbeler edition 14 p 140. Sketch Problems 4-22/23 with F = 100 N .

• Maximum moment is 34.42 N m when θ = 154.2◦ or θ = 334.2◦ .

• Minimum moment is 0 N m when θ = 64.2◦ or θ = 244.2◦ .

45
LECTURE UNIT 2.2

3 (a) Hibbeler edition 14 p 152. Problem 4.54.

(i) M A = 80i + 400j − 50k N m.


(ii) M C = −80i + 240j − 110k N m.


(iii) M AC = −160i + 160j N m.


(iv) M AC = −160i + 160j N m.
(v) Both answers are the same. This complies with the theory.
(vi) (a) M AC is the projection of M A on the axis through A and C.
(b) M AC is also the projection of M C on the axis through A and C.
(vii) The moments of F about the three coordinate axes are
M x = −80i N m, M y = 400j N m and M z = −150k N m.


• M y = M A · j j = 400j N m since A lies on the y−axis and


• M x = M C · i i = −80i N m since C lies on the x−axis.

LECTURE UNIT 2.3

2 (b) Hibbeler edition 14 p 165. Problem 4.96.

If the body is rigid and in equilibrium, the resultant couple moment can act
anywhere on the body.

LECTURE UNIT 2.4

3 (c) Hibbeler edition 14 p 174. Problem 4.101.

(iii) Equivalent systems at A and B.

B

4.462 kN m B

69.9◦

A 8.274 kN
9.769 kN m
69.9◦

8.274 kN A
(iv) Similarities: Resultant forces in both cases have the same magnitudes,
directions and senses.
Differences: Resultant forces act at different points. Moments have differ-
ent magnitudes and directions and senses.

46
LECTURE UNIT 2.8

4. The simplest equivalent system is a wrench with force F w = −40i N and couple
moment M w = −30i N m.
The line of action of the wrench is described by the cartesian equations y = −1
and z = 0.
This line intersects the pipe assembly nowhere, hence the given force system
can’t be replaced by a wrench acting on the pipe assembly.

LECTURE UNIT 4.1

3. Centroid (x, y, z) = (−2.035, 6.513, 4.885) mm.

4. Center of gravity (x, y) = (1.594, 9.126) m.

5. Centroid (y, z) = (4.907, 2.484) m.

6. Centroid (x, y) = (1.122, −0.2255) cm.

LECTURE UNIT 4.3

3. Ix = 0.1474r4 and Iy = 7.136 × 10−3 r4 .

47
 TEXTBOOK ANSWERS

The answers of the following problems are not given in the back of the textbook.

Note: The answers are not rounded to the required precision as stipulated in Re-
marks # 3 and 4 in Lecture Unit 1.1.

1-8 (a) kN m (b) Gg/m (c) µN/s2 (d) GN/s

1-12 (a) 58.3 km (b) 68.5 s (c) 2.55 kN (d) 7.56 M g

2-16 θ = 54.9◦ , FR = 10.4 kN .

2-40 FR = 546, kN , θ = 252.6◦ .

2-68 F3 = 428 N , α = 88.3◦ , β = 20.6◦ , γ = 69.5◦ .

2-80 F 1 = 260i − 150k) N , F 2 = 177i + 306j − 354k) N .

2-96 FR = 194 N , α = 90.9◦ , β = 76.4◦ , γ = 166.4◦ .

3-16 FAB = FAC = 2.45 cos θ kN , shortest length is 1.72 m.

3-24 mB = 3.58 kg, normal force = 19.7 N .

3-28 FBC = 2.90 kN , y = −841 mm.

3-32 Maximum mass is 20.3 kg.

4-12 Moment of F 1 about point B is 150 N m anticlockwise.

Moment of F 2 about point B is 600 N m anticlockwise.
Moment of F 3 about point B is 0 N m.


4-44 10.6i + 13.1j + 29.2k N m.


4-48 1.56i − 0.750j − 1.00k kN m.

4-96 F1 = 87.2 N , F2 = 112 N , F3 = 100 N .

4-140 Equivalent resultant force has magnitude 70 N and acts 0.107 m to the right of
O.

4-156 Equivalent resultant force has magnitude 51 N and acts 17.9 m to the right of
O.

5-16 T = 0.5W sin θ.

5-68 RD = RE = 113 N , RF = 68.7 N .

5-80 TB = 16.7 kN , Ax = 0, Ay = 5.00], kN , Az = 16.7 kN

48
6.68 Normal force between collar and smooth rod is 3.67 kN . Moment that develops
between collar and rod is 5.55 kN m.
Cx = 2.89 kN and Cy = 1.32 kN .

9-56 x = 24.4 mm, y = 40.6 mm.

49
CLICKUP EVALUATION
Note: The information in this section is relevant for the lecture assignments (LA),
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tails for the class tests, semester tests and examination will be communicated closer
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preparation for the tutorials (TP) are simple and basic questions to test the
student’s understanding of the principles and fundamental concepts covered in
the module.
3. The questions in the (iii) worksheets (TW) serve to prepare the student for the
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8. Students get: one (1) attempt per TW evaluation. Please note the following
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your answer will be different to your fellow students’ answers.
• You will be able to start the tutorial between 00:01 on a given Tuesday
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is due.
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started) to complete the tutorial. This is total time and includes the time
you are not actively working on the tutorial. Note all attempts will close
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tant that you submit your answers as you enter them. You can however
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exit the browser and log in again at a later stage.

50
 • You will be able to see whether you have the answer correct or incorrect.
If your answer is incorrect, you can then improve your answer and submit
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11. You will be required to enter your answers in provided answer blocks you will


see a block in which you must enter your answer .

12. Take note of the following points when entering your answers:

(a) The number of significant digits or decimal digits are important (see the
remarks in Lecture Unit 1.1).
(i) All answers, except angles, must be rounded to 4 significant digits.
The scalar 32 114 will be represented as 32 110.
√
32 110 32 114 ×

The scalar 63.73 stays 63.73.

√
63.73

51
 The scalar 0.0020416 will be represented as 0.002042.
√
0.002042 0.0020416 ×
(ii) All angles must be rounded to 1 decimal place. The answer block will
be given in the format ◦

Do not include the degree symbol in the answer block.

The angle 71.26◦ will be represented as 71.3◦ .

◦ √ ◦
71.3 71.26 ×

The angle 213.4◦ stays 213.4◦ .

◦ √
213.4
(b) Do not include units in your answer blocks. Represent your answer in
terms of the units as indicated next to the answer block.
For instance, if you calculated an answer F = 51.73 N , you have to fill in
only the scalar 51.73 and NOT the unit N .
The answer block will be presented as N
√
51.73 N 51.73 N N ×

(c) Decimal point notation.

(i) Use the decimal point notation (use a period “.”) in the representation
of numbers, and not the decimal comma notation “,” .
If F = 51.73 N , then
√
51.73 N 51,73 N ×

(ii) Do not use scientific notation. If m = 321.76 kg, the value, rounded
to four significant digits is m = 321.8 kg.
√
321.8 kg 3.218 × 102 kg ×

(d) Cartesian vector format.

Answers in cartesian vector format are represented with three blocks.
(i) The first block represents the i−component,
(ii) the second block represents the j−component and
(iii) the third block represents the k−component.
(e) Enter only the magnitude and sign of the component.
(f) Omit i, j and k in the blocks.
(g) Omit units in the blocks.

52
 Consider for example the position vector (2i − 1k) m. The answer block will be
given in the format
 
i + j + k m

You will enter the answer as

  √
2 i + 0 j + −1 k m

The following representations are wrong.

 
• 2i i + 0j j + −1k k m ×
 
• 2m i + 0m j + −1m k m ×
 
• 2i m i + 0j m j + −1k m k m ×

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¯
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18. Any further instructions will be communicated via clickUP.

53
 Geometric properties of line and area elements

πr
y L= L = πr
2
L = 2θr 2r
r
π
θ C
b
x C C
θ r b b
r
r sin θ
θ

Circular arc segment Quarter and semicircle arcs

πr2
A = θr2 y A=
y 4

r4 r πr4
r


Ix = θ − 0.5 sin 2θ b Ix =
4 C 16
θ C x
b
x
θ πr4
r4

Iy = θ + 0.5 sin 2θ 4r Iy =
4 16
3π
2r sin θ
3θ Circular sector area Quarter circle area

πr2
y A= y A = πr2 πr4
2 Ix =
4
πr4
C r b Ix =
8
r πr4
C Iy =
x b
x 4
4r πr4
Iy =
3π 8

Semicircular area Circular area

y bh
A = bh bh3 A=
Ix = 2
C 12 bh3
h b
x h C Ix =
b
x 36
b3 h
Iy =
b 12 b
h
3
Rectangular area Triangular area